A condition for finite dimensional convexity

Abstract

Here RN is N-dimensional Euclidean space. We use Cech homology with integer coefficient group [1]. Plane will be used for N-1 dimensional hyperplane. A support plane, P, is a plane meeting K where K is in just one of the two closed half spaces determined by P. KflP is the support set for P. A weaker theorem, requiring the added hypothesis of local contractibility for K was proved by Kuhn [2 ] who pointed out this hypothesis was essential for any proof based on his methods. Liberman [3] has established a slightly weaker theorem requiring contractibility for the support sets as the culmination of a long chain of subsidiary results. PROOF. We need Rn, the linear extension of K, so n 1. Let Sn-' and Y refer to the n-1 topological sphere and to the metric unit n -1 sphere respectively. Each point y in Y determines a unique supporting hyperplane P2, whose normals pointing away from K are parallel translates in Rn of the vector oy. We write X for the set of boundary points of K and Xv for the support set P/nX. Let